On Shadow Boundaries of Centrally Symmetric Convex Bodies

We discuss the concept of the so-called shadow boundary belonging to a given direction x of Euclidean n-space R lying in the boundary of a centrally symmetric convex body K. Actually, K can be considered as the unit ball of a finite dimensional normed linear (= Minkowski) space. We introduce the notion of the general parameter spheres of K corresponding to the above direction x and prove that if all of the non-degenerate general parameter spheres are topological manifolds, then the shadow boundary itself becomes a topological manifold as well. Moreover, using the approximation theorem of cell-like maps we obtain that all these parameter spheres are homeomorphic to the (n− 2)-dimensional sphere S(n−2). We also prove that the bisector (i.e., the equidistant set with respect to the norm) belonging to the direction x is homeomorphic to R(n−1) iff all of the non-degenerate general parameter spheres are (n− 2)-manifolds. This implies that if the bisector is a homeomorphic copy of R(n−1), then the corresponding shadow boundary is a topological (n− 2)-sphere. MSC 2000: 52A21, 52A10, 46C15